Properties

Label 2-323400-1.1-c1-0-118
Degree $2$
Conductor $323400$
Sign $-1$
Analytic cond. $2582.36$
Root an. cond. $50.8169$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 11-s − 4·13-s + 2·17-s − 4·19-s − 2·23-s − 27-s + 2·31-s − 33-s − 8·37-s + 4·39-s + 6·41-s + 4·43-s − 2·51-s − 6·53-s + 4·57-s − 4·59-s − 6·61-s + 4·67-s + 2·69-s + 4·73-s − 14·79-s + 81-s + 6·83-s − 2·93-s + 10·97-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s − 0.192·27-s + 0.359·31-s − 0.174·33-s − 1.31·37-s + 0.640·39-s + 0.937·41-s + 0.609·43-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.488·67-s + 0.240·69-s + 0.468·73-s − 1.57·79-s + 1/9·81-s + 0.658·83-s − 0.207·93-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(323400\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(2582.36\)
Root analytic conductor: \(50.8169\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{323400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 323400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.57868607118018, −12.49490043652559, −11.95450906022750, −11.56055670741832, −11.00162855555151, −10.57735558640393, −10.14901916454287, −9.753973607404307, −9.210456922721887, −8.804720454524982, −8.181508568434353, −7.662237792180708, −7.331802792322960, −6.728004660749290, −6.306930917919271, −5.841067997925315, −5.314363207491435, −4.767954304105114, −4.398939823718929, −3.828947332328183, −3.212154165030946, −2.578746304500326, −2.003961920761185, −1.443413314204069, −0.6371641105111448, 0, 0.6371641105111448, 1.443413314204069, 2.003961920761185, 2.578746304500326, 3.212154165030946, 3.828947332328183, 4.398939823718929, 4.767954304105114, 5.314363207491435, 5.841067997925315, 6.306930917919271, 6.728004660749290, 7.331802792322960, 7.662237792180708, 8.181508568434353, 8.804720454524982, 9.210456922721887, 9.753973607404307, 10.14901916454287, 10.57735558640393, 11.00162855555151, 11.56055670741832, 11.95450906022750, 12.49490043652559, 12.57868607118018

Graph of the $Z$-function along the critical line